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PROFESSOR: So you may recall
that, in the last lecture, we

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00:00:25,090 --> 00:00:29,220
more or less solved the drunken
student problem,

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looking at a random walk.

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I now want to move on and
discuss some variants of the

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random walk problem that are
collectively known as biased

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random walks.

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So the notion here is, the walk
is still stochastic but

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there is some bias in the
direction, so the movements

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00:01:01,230 --> 00:01:05,500
are not uniformly distributed or
equally distributed in all

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directions.

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As I go through this, I want
you to sort of, think in

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advance about what the take-home
message should be.

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One, and this is probably the
most important part for today,

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is I want to illustrate how by
designing our programs in a

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nice way around classes, we
can change them to do

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00:01:29,670 --> 00:01:35,430
something else writing a
minimal amount of code.

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00:01:35,430 --> 00:01:39,180
And the idea here is that
classes are not there just to

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provide some syntax that makes
life complicated, but to

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actually provide a mechanism
that lets us structure our

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00:01:46,050 --> 00:01:52,380
programs in a way that we
can modify them later.

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00:01:52,380 --> 00:01:55,670
I also want to give you a bit
more experience looking at

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00:01:55,670 --> 00:01:59,820
data, and understanding how,
when you get the results of a

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00:01:59,820 --> 00:02:04,000
simulation or anything else that
provides you with data,

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00:02:04,000 --> 00:02:08,720
you can plot, them study the
plots, and try and use those

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00:02:08,720 --> 00:02:12,460
to develop an understanding
about what's going on.

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00:02:12,460 --> 00:02:16,860
And finally, I want to just get
you thinking more about

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00:02:16,860 --> 00:02:18,560
random behavior.

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00:02:18,560 --> 00:02:21,080
Because we're going to talk a
lot more this semester and in

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general throughout your careers,
you'll see a lot of

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uses of stochastic things.

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00:02:26,530 --> 00:02:31,280
All right, so let's think about
these biased walks.

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00:02:31,280 --> 00:02:36,610
So for example, assume that
the drunk grew up in South

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Florida, and really hates
the New England winter.

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Well there might be a bias that
even though the drunk is

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not in total control, she's kind
of wandering southward.

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So there's a bias when taking a
step to more likely go south

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00:02:52,380 --> 00:02:55,400
than certainly north.

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00:02:55,400 --> 00:02:59,160
Or imagine that the drunk is
photosensitive and moves

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either towards the sun
or away from the sun,

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00:03:02,030 --> 00:03:04,410
or some such thing.

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00:03:04,410 --> 00:03:06,910
So we're going to look at
different things of that

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00:03:06,910 --> 00:03:09,970
nature and think about, I'll
come back to Chutes and

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Ladders later in the lecture,
you can speculate

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on why this is here.

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And we'll look at that.

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So, this is on the front page
of your handout here.

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I've taken the random walk
program we looked at last time

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and changed it a bit.

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And in particular,
I've changed the

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way the drunk works.

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So let's look at the code.

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Wow, haven't even done
anything it's --

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see how this works.

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So now, drunk itself is a very
small class, one I don't

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actually intend to
ever instantiate.

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I've pared it down, it has
only two methods in it.

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It has an init, which is the
same as before, and a move

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which is the same as before,
almost the same.

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Now I'm going to have the
class usual drunk.

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That's the drunk we looked
at the last time.

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And this usual drunk will behave
just like the drunk we

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looked at last time.

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In here, I'm going to override
the move method of the

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inherited superclass, and what
this is going to do is, it's

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going to make the same random
choice it made before and then

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it's going to call drunk
dot move with

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the new compass point.

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So all I've done is, I've taken
a couple of lines of

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code out of the previous
implementation and moved it to

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the subclass.

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So it's now, excuse me, in the
subclass I'm making the

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decision about which
direction to move.

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Once I've made that decision,
then I can call the drunk in

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the superclass.

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Notice here, when I've made
the call, drunk dot move.

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So usually you're expecting
me to write

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something like d dot move.

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The object.

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Here, instead I'm specifying
which class to get the move

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00:06:01,170 --> 00:06:07,710
from, So this is saying, don't
call my local move, call the

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move from the superclass.

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There are a number of other
syntaxes I could've used to

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accomplish the same thing.

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I used this because it seemed
the most straightforward.

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00:06:24,830 --> 00:06:29,960
Now let's compare the usual
drunk to the cold drunk.

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So the cold drunk, again, I've
overridden the move, and now

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00:06:35,560 --> 00:06:37,810
it does something different.

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00:06:37,810 --> 00:06:42,730
It gets the random compass point
as before, but if it

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00:06:42,730 --> 00:06:47,420
happens to be south, it
calls drunk dot move

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00:06:47,420 --> 00:06:53,500
with twice the distance.

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00:06:53,500 --> 00:06:59,070
So whenever I get the notion
I'm moving south, because I

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00:06:59,070 --> 00:07:04,210
love the warmth, I actually move
twice as fast. So instead

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00:07:04,210 --> 00:07:07,100
of taking one step, which I do
to the other compass points,

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00:07:07,100 --> 00:07:11,360
if I happen to be going south
I'll take two steps.

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00:07:11,360 --> 00:07:17,830
So I'm biased towards
moving southward.

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00:07:17,830 --> 00:07:23,640
And then I call drunk
dot move.

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00:07:23,640 --> 00:07:26,340
Otherwise, I call drunk
dot move with

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00:07:26,340 --> 00:07:30,540
the distances before.

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So I've isolated into the
subclass this notion of being

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heat-seeking.

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We've got a third sub class,
ew drunk, known

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00:07:42,170 --> 00:07:47,100
for east west drunk.

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00:07:47,100 --> 00:07:52,160
And here, what I've done is, I
choose to compass point and

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then while it's not either east
or west, I choose again.

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00:07:57,200 --> 00:08:00,790
So this drunk can only move
eastward or westward.

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00:08:00,790 --> 00:08:07,810
Can't move north or south.

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00:08:07,810 --> 00:08:12,790
So again you'll see I'm able to
have three kinds of drunks.

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00:08:12,790 --> 00:08:15,550
Who as we will see, exhibit
rather different behavior from

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00:08:15,550 --> 00:08:25,650
each other with a minimal amount
of change to the code.

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00:08:25,650 --> 00:08:33,880
If we look down at the bottom,
we'll see that the code is

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00:08:33,880 --> 00:08:37,540
pretty much exactly what
it was last time.

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00:08:37,540 --> 00:08:39,340
By the way, this doesn't
quite match the

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00:08:39,340 --> 00:08:40,970
code in your handout.

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I decided to make the plots
a little bit prettier with

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better titles and things, but
when I tried to put these in

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00:08:47,550 --> 00:08:49,740
your handouts it wouldn't fit.

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00:08:49,740 --> 00:08:53,360
So I have a little more compact
things, but this is, I

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00:08:53,360 --> 00:08:55,630
think, a better way
to write it.

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00:08:55,630 --> 00:09:00,520
But essentially, perform trial,
perform sim, answer

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quest, are all identical to
what they were before.

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00:09:05,940 --> 00:09:11,850
So because I structured my
program in such a way that the

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00:09:11,850 --> 00:09:15,620
drunk's behavior was independent
of all the code

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00:09:15,620 --> 00:09:21,420
for the generating the trials
and plotting the results, I

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didn't have to change
any of that.

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And this is the key
to good design.

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Is to isolate decisions in small
parts of your program,

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00:09:33,270 --> 00:09:37,650
so when as inevitable happens,
you want to enhance the

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00:09:37,650 --> 00:09:40,860
program, or you discover you've
done something wrong,

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00:09:40,860 --> 00:09:48,100
the number changes you
make is not large.

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00:09:48,100 --> 00:09:53,290
OK, one thing we do need to look
at down here is, it's not

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00:09:53,290 --> 00:09:57,550
quite the same as before,
because we'll see that there's

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00:09:57,550 --> 00:10:06,240
an extra parameter to these
things called drunk type.

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00:10:06,240 --> 00:10:10,940
So if we start with answer
question, it takes as before,

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00:10:10,940 --> 00:10:14,700
the time, the number of trials,
a title, and this one

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00:10:14,700 --> 00:10:18,960
extra thing called
the drunk type.

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00:10:18,960 --> 00:10:22,620
And then when it goes on and
it calls perform sim, it

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00:10:22,620 --> 00:10:27,310
passes the drunk type.

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00:10:27,310 --> 00:10:38,710
And this is the key line here.

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00:10:38,710 --> 00:10:41,690
Before, you may recall
we just said d equals

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00:10:41,690 --> 00:10:45,340
drunk with the arguments.

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00:10:45,340 --> 00:10:51,730
Here, I say d equals
drunk type.

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00:10:51,730 --> 00:10:56,650
So what is drunk type?

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00:10:56,650 --> 00:11:00,710
Drunk type is itself a type.

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00:11:00,710 --> 00:11:04,350
We can use types, you know
variables can have as their

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00:11:04,350 --> 00:11:09,610
values types, parameters can
have as their values types.

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00:11:09,610 --> 00:11:13,630
So I'm just passing the type
around so that here, when it

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comes time to get a drunk,
I get a drunk of

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00:11:17,160 --> 00:11:21,620
the appropriate type.

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00:11:21,620 --> 00:11:25,110
So I can now answer the question
about a usual drunk,

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00:11:25,110 --> 00:11:31,700
or a cold drunk,
or a ew drunk.

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00:11:31,700 --> 00:11:36,880
This is very common kind of
programming paradigm.

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00:11:36,880 --> 00:11:40,430
We've talked about it before,
but this is yet another

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example of how we use
polymorphism.

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00:11:44,500 --> 00:11:47,950
Polymorphism.

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00:11:47,950 --> 00:11:51,470
We write our code in such a way
that it works with many

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00:11:51,470 --> 00:11:54,950
types of objects.

164
00:11:54,950 --> 00:11:57,740
Here we're writing code
that will work with

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00:11:57,740 --> 00:12:01,630
any subtype of drunk.

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00:12:01,630 --> 00:12:04,170
Now I'd better not
pass it a float.

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00:12:04,170 --> 00:12:08,230
You know, if I answer quest, and
instead of something like

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00:12:08,230 --> 00:12:12,360
usual drunk or cold drunk or e
w drunk, I pass it float or

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00:12:12,360 --> 00:12:17,560
int, the world will come
crashing down around my head.

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00:12:17,560 --> 00:12:21,720
But I haven't, so you'll see
down here, I pass it num

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00:12:21,720 --> 00:12:27,130
steps, number trials,
and usual drunk.

172
00:12:27,130 --> 00:12:28,530
Which is the name of?

173
00:12:28,530 --> 00:12:33,050
A class.

174
00:12:33,050 --> 00:12:38,760
Any questions about this?

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00:12:38,760 --> 00:12:39,160
Yes?

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00:12:39,160 --> 00:12:45,360
STUDENT: [INAUDIBLE]

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00:12:45,360 --> 00:12:46,330
PROFESSOR: Louder, please?

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00:12:46,330 --> 00:12:53,370
STUDENT: [INAUDIBLE]

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00:12:53,370 --> 00:12:54,950
PROFESSOR: Good,
good question.

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00:12:54,950 --> 00:12:58,250
The question is, does a drunk
type make a new drunk?

181
00:12:58,250 --> 00:13:00,630
Now, I have so much candy to
choose from, I don't know what

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00:13:00,630 --> 00:13:02,470
to throw you.

183
00:13:02,470 --> 00:13:07,670
Do you like Tootsie Rolls,
Snickers, Tootsie Roll Pops,

184
00:13:07,670 --> 00:13:11,220
Dots, what's your choice?

185
00:13:11,220 --> 00:13:19,370
Snickers.

186
00:13:19,370 --> 00:13:23,040
OK, so what is it?

187
00:13:23,040 --> 00:13:29,970
Down here nothing is created
when that call was made.

188
00:13:29,970 --> 00:13:34,180
It was if I had typed the
word float or int.

189
00:13:34,180 --> 00:13:41,460
I'm merely saying, here is a
type, and the parameter should

190
00:13:41,460 --> 00:13:45,860
now be bound to that type.

191
00:13:45,860 --> 00:13:49,680
Now any time you use the
parameter, it's just as if you

192
00:13:49,680 --> 00:13:56,240
had written the string, in
this case, usual drunk.

193
00:13:56,240 --> 00:14:08,790
So when I get up to here, think
about, it will take, the

194
00:14:08,790 --> 00:14:19,610
variable drunk type, evaluate
it, it will evaluate to a type

195
00:14:19,610 --> 00:14:23,040
usual drunk, say, in
the first call.

196
00:14:23,040 --> 00:14:28,040
And then it will, just as if
I had written usual drunk,

197
00:14:28,040 --> 00:14:34,430
invoke the init method of drunk,
and create a new drunk,

198
00:14:34,430 --> 00:14:36,890
but this drunk will be
of type usual drunk

199
00:14:36,890 --> 00:14:40,300
rather than a drunk.

200
00:14:40,300 --> 00:14:44,970
Does that make sense?

201
00:14:44,970 --> 00:14:50,320
All right, so this avoids my
having to write a whole bunch

202
00:14:50,320 --> 00:14:54,590
of tests, I could have passed
a string, and said if the

203
00:14:54,590 --> 00:14:59,140
string said usual drunk, then
d equals something.

204
00:14:59,140 --> 00:15:00,860
But here instead of
that, I'm actually

205
00:15:00,860 --> 00:15:07,330
passing the type itself.

206
00:15:07,330 --> 00:15:12,710
OK?

207
00:15:12,710 --> 00:15:16,300
So other than the fact that I
have to pass the type around,

208
00:15:16,300 --> 00:15:21,570
this is exactly the
same as before.

209
00:15:21,570 --> 00:15:26,150
Now just for fun, we saw that
last time when I asked you

210
00:15:26,150 --> 00:15:30,100
what did usual drunk do, there
was some confusion.

211
00:15:30,100 --> 00:15:34,410
Not confusion, just a bad guess
about how the drunk

212
00:15:34,410 --> 00:15:35,950
would wander.

213
00:15:35,950 --> 00:15:38,950
So, let's try it again.

214
00:15:38,950 --> 00:15:44,490
What do you think about
the cold drunk?

215
00:15:44,490 --> 00:15:48,850
Will the cold drunk get further
from the origin than

216
00:15:48,850 --> 00:15:50,740
the usual drunk?

217
00:15:50,740 --> 00:15:51,880
Not further from the origin?

218
00:15:51,880 --> 00:15:55,450
Who thinks the cold drunk
will be further away

219
00:15:55,450 --> 00:15:56,710
the end of 500 steps?

220
00:15:56,710 --> 00:16:01,610
Who thinks the cold drunk will
not be further away?

221
00:16:01,610 --> 00:16:05,590
And I'll bet we can all agr --
got that right, I think -- and

222
00:16:05,590 --> 00:16:08,470
I'll bet you'll all agree that,
with higher probability,

223
00:16:08,470 --> 00:16:12,870
the cold drunk will be south
of the origin than north of

224
00:16:12,870 --> 00:16:14,520
the origin.

225
00:16:14,520 --> 00:16:19,440
More interesting, how
about the ew drunk?

226
00:16:19,440 --> 00:16:24,120
So this is a drunk that only
goes east or west, east or

227
00:16:24,120 --> 00:16:31,950
west. Will this drunk, well,
first of all, what's the

228
00:16:31,950 --> 00:16:33,280
expected distance?

229
00:16:33,280 --> 00:16:34,880
Do you think it should
be close to zero or

230
00:16:34,880 --> 00:16:37,560
not close to zero?

231
00:16:37,560 --> 00:16:41,180
Who thinks it should
be close to 0?

232
00:16:41,180 --> 00:16:44,780
Who thinks it won't
be close to 0?

233
00:16:44,780 --> 00:16:47,370
Well you guys have all fallen
for the same trick you fell

234
00:16:47,370 --> 00:16:49,530
for last time.

235
00:16:49,530 --> 00:16:50,510
Fooled you twice.

236
00:16:50,510 --> 00:16:54,230
Be careful, one more
and you're out.

237
00:16:54,230 --> 00:16:56,560
As we'll see.

238
00:16:56,560 --> 00:16:57,770
But it's interesting.

239
00:16:57,770 --> 00:17:12,680
So let's run this program
and see what happens.

240
00:17:12,680 --> 00:17:15,590
I can take a just a second
or two and we should

241
00:17:15,590 --> 00:17:22,470
get a bunch of plots.

242
00:17:22,470 --> 00:17:25,710
All right, so here's
the usual drunk.

243
00:17:25,710 --> 00:17:33,050
100 trials and just as before,
wandered away.

244
00:17:33,050 --> 00:17:36,440
Moderately smooth, but of course
little ups and downs,

245
00:17:36,440 --> 00:17:39,990
maybe suggesting averaging
over 100 trials

246
00:17:39,990 --> 00:17:42,230
is not quite enough.

247
00:17:42,230 --> 00:17:45,920
I felt a little better if it
were smoother, but the trend

248
00:17:45,920 --> 00:17:52,970
is pretty clear here, I think
we probably don't doubt it.

249
00:17:52,970 --> 00:17:56,530
Well, look at cold drunk.

250
00:17:56,530 --> 00:18:00,520
Like a shot heading south.

251
00:18:00,520 --> 00:18:04,930
So here we see, there's bias,
that if you get south you go

252
00:18:04,930 --> 00:18:08,670
twice as far makes a
huge difference.

253
00:18:08,670 --> 00:18:13,450
Instead of being roughly 20
steps away, it's roughly 120

254
00:18:13,450 --> 00:18:16,550
steps away.

255
00:18:16,550 --> 00:18:21,500
So we see, and this is something
to think about, a

256
00:18:21,500 --> 00:18:27,520
small bias can end up making a
huge difference over time.

257
00:18:27,520 --> 00:18:31,230
This is why casinos
do so well.

258
00:18:31,230 --> 00:18:34,370
They don't have to have a very
much of a bias in their favor

259
00:18:34,370 --> 00:18:37,640
on each roll of the dice in
order to accumulate a lot of

260
00:18:37,640 --> 00:18:40,410
money over time.

261
00:18:40,410 --> 00:18:43,040
It's why if you just weigh your
dice a little bit, so

262
00:18:43,040 --> 00:18:47,020
that they're not fair, you'll
do very very well.

263
00:18:47,020 --> 00:18:52,100
I'm not recommending that.

264
00:18:52,100 --> 00:18:55,220
Kind of interesting.

265
00:18:55,220 --> 00:18:59,680
The e w drunk is pretty much the
same distance away as the

266
00:18:59,680 --> 00:19:04,440
usual drunk.

267
00:19:04,440 --> 00:19:10,090
This is why we run these
simulations, is because we

268
00:19:10,090 --> 00:19:10,780
learn things.

269
00:19:10,780 --> 00:19:13,920
Now, I have to confess,
it fooled me, too.

270
00:19:13,920 --> 00:19:16,570
The first time I ran this, I
looked at this and said, wait

271
00:19:16,570 --> 00:19:20,800
a minute, that's not
what I expected.

272
00:19:20,800 --> 00:19:24,610
So I scratched my head, and
thought about, all right, why

273
00:19:24,610 --> 00:19:27,770
is it behaving this way?

274
00:19:27,770 --> 00:19:32,260
This happens me a lot in my own
research, that I crunch

275
00:19:32,260 --> 00:19:36,950
some data, do some plots, look
at it, get surprised, and on

276
00:19:36,950 --> 00:19:41,100
the basis of that I learned
something about the underlying

277
00:19:41,100 --> 00:19:42,960
system I'm modeling.

278
00:19:42,960 --> 00:19:46,910
Because it provokes me to
think in ways I had not

279
00:19:46,910 --> 00:19:49,510
thought before I saw the data.

280
00:19:49,510 --> 00:19:50,900
This is an important lesson.

281
00:19:50,900 --> 00:19:54,740
You don't ignore the data, you
try and explain the data.

282
00:19:54,740 --> 00:19:58,200
So when I thought about what
happened, I realize that while

283
00:19:58,200 --> 00:20:02,290
I might have started at the
origin, and after the first

284
00:20:02,290 --> 00:20:07,950
step I'm equally likely
to be here or here.

285
00:20:07,950 --> 00:20:17,260
Once I've taken the first step,
now my expected location

286
00:20:17,260 --> 00:20:20,600
is not zero, right?

287
00:20:20,600 --> 00:20:25,580
The first step, my expected
location was zero.

288
00:20:25,580 --> 00:20:29,550
Because equally likely to
be plus 1 or minus 1.

289
00:20:29,550 --> 00:20:34,810
But once I've taken a step and
I happen to be here, my

290
00:20:34,810 --> 00:20:40,820
expected location, if I'm here,
is plus 1, it's not 0.

291
00:20:40,820 --> 00:20:43,630
Because half the time I'll get
back to 0, but half the time

292
00:20:43,630 --> 00:20:48,100
I'll get to 2.

293
00:20:48,100 --> 00:20:55,100
So once I have moved away from
where I've started, the

294
00:20:55,100 --> 00:21:03,160
expectation of where I'll
end up has changed.

295
00:21:03,160 --> 00:21:07,880
This is kind of a profound thing
to think about, because

296
00:21:07,880 --> 00:21:12,090
when we look at it, this kind of
behavior explains a lot of

297
00:21:12,090 --> 00:21:14,820
surprising results in life.

298
00:21:14,820 --> 00:21:21,620
Where, if you get a small run of
random events that take you

299
00:21:21,620 --> 00:21:28,170
far from where you started,
it's hard to get back.

300
00:21:28,170 --> 00:21:32,160
So if you think about the stock
market, for example, if

301
00:21:32,160 --> 00:21:36,420
you happen to get unlucky and
have a few days or weeks or

302
00:21:36,420 --> 00:21:40,490
months, as the case may be,
where you get a run of bad

303
00:21:40,490 --> 00:21:44,910
days, it's really hard to get
back if all the movements

304
00:21:44,910 --> 00:21:46,770
after that are random.

305
00:21:46,770 --> 00:21:50,710
Because you've established
a new baseline.

306
00:21:50,710 --> 00:21:53,240
And as we'll see later, when
you start doing things at

307
00:21:53,240 --> 00:21:57,740
random, it's likely you'll get a
few runs in one direction or

308
00:21:57,740 --> 00:22:01,940
another, and then you've got
a new baseline, and it's

309
00:22:01,940 --> 00:22:04,960
unlikely to get back to where
you were originally.

310
00:22:04,960 --> 00:22:07,200
We'll see a lot of
examples of this.

311
00:22:07,200 --> 00:22:13,530
But that's what's happening
here, and that's why it looks

312
00:22:13,530 --> 00:22:17,140
the way it does.

313
00:22:17,140 --> 00:22:20,450
That makes sense to people?

314
00:22:20,450 --> 00:22:24,430
Well, so I had this theory,
as I often do when I have

315
00:22:24,430 --> 00:22:28,650
theories, I don't believe them,
so I decided I better

316
00:22:28,650 --> 00:22:34,120
write some more code
to check it out.

317
00:22:34,120 --> 00:22:40,020
So, what I did is, I took the
previous simulation and now

318
00:22:40,020 --> 00:22:43,330
instead of looking at different
kinds of drunks, I

319
00:22:43,330 --> 00:22:46,990
got more aggressive in
thinking about my

320
00:22:46,990 --> 00:22:50,440
analysis of the data.

321
00:22:50,440 --> 00:22:53,590
So again, there's
a lesson here.

322
00:22:53,590 --> 00:22:58,900
I decided I wanted to analyze
my data in new ways.

323
00:22:58,900 --> 00:23:03,160
And the beauty of it was, I
could do that without changing

324
00:23:03,160 --> 00:23:07,912
either any of the code about the
drunk, or any they of the

325
00:23:07,912 --> 00:23:10,970
code about performing the
simulation, really.

326
00:23:10,970 --> 00:23:14,890
All I had to do was do a
different kind of analysis to

327
00:23:14,890 --> 00:23:18,250
a first approximation.

328
00:23:18,250 --> 00:23:26,170
So, let's look at what
I've done here.

329
00:23:26,170 --> 00:23:30,990
What I've decided to do here is
not only plot, and again, I

330
00:23:30,990 --> 00:23:37,830
think you'll see this code, not
only plot how far from the

331
00:23:37,830 --> 00:23:44,300
origin the drunk is over time,
but I'm going to actually plot

332
00:23:44,300 --> 00:23:46,870
two other things.

333
00:23:46,870 --> 00:23:51,880
I'm going to use a scatter plot,
I think I use scatter in

334
00:23:51,880 --> 00:23:54,870
this one, I do, to show
where the drunk is at

335
00:23:54,870 --> 00:23:56,070
each period of time.

336
00:23:56,070 --> 00:23:58,700
Actually, just all the
places the drunk

337
00:23:58,700 --> 00:24:00,880
ends up in each trial.

338
00:24:00,880 --> 00:24:04,980
This will give me a visualized
way just see, OK, in my, say,

339
00:24:04,980 --> 00:24:08,860
100 trials, how many the drunk
ended up near the origin, how

340
00:24:08,860 --> 00:24:11,160
many of them ended up
really far away?

341
00:24:11,160 --> 00:24:16,240
And just, allow me to visualize
what's going on.

342
00:24:16,240 --> 00:24:19,390
And then I'm going to also use
a histogram to sort of

343
00:24:19,390 --> 00:24:21,400
summarize some things.

344
00:24:21,400 --> 00:24:34,430
To get a look at the
distribution of the data.

345
00:24:34,430 --> 00:24:38,620
We've been just looking at
average results, but it's

346
00:24:38,620 --> 00:24:41,610
often the case that the average
does not really tell

347
00:24:41,610 --> 00:24:44,190
the whole story.

348
00:24:44,190 --> 00:24:46,490
You want to ask, well, how many
of the drunks ended up

349
00:24:46,490 --> 00:24:48,930
really far away?

350
00:24:48,930 --> 00:24:51,830
How many of the drunks ended
up where they started?

351
00:24:51,830 --> 00:24:55,780
You can't tell that
from the averages.

352
00:24:55,780 --> 00:24:59,860
So you use a distribution, which
tells you information

353
00:24:59,860 --> 00:25:03,500
about how many people
ended up where.

354
00:25:03,500 --> 00:25:07,400
So we again saw that, in a less
pleasant context, maybe,

355
00:25:07,400 --> 00:25:10,910
in looking at the quizzes,
remember I sent you some plots

356
00:25:10,910 --> 00:25:13,470
about the quizzes, and
I didn't just say,

357
00:25:13,470 --> 00:25:16,320
here was the average.

358
00:25:16,320 --> 00:25:20,750
I showed you the distribution
of grades.

359
00:25:20,750 --> 00:25:23,970
That allowed you to see, not
only where you were relative

360
00:25:23,970 --> 00:25:27,360
to the average, but whether
you were an outlier.

361
00:25:27,360 --> 00:25:30,160
Was your grade much better
than the average?

362
00:25:30,160 --> 00:25:33,980
Or were there a whole bunch of
people around where you are?

363
00:25:33,980 --> 00:25:37,090
So there's a lot of information
there, that will

364
00:25:37,090 --> 00:25:41,280
help us understand what's
really going on.

365
00:25:41,280 --> 00:25:45,940
So we'll run this now.

366
00:25:45,940 --> 00:25:48,440
Again, there's nothing very
interesting here about how I

367
00:25:48,440 --> 00:25:52,200
did it, I just used some of
these Pylab dot hist, which

368
00:25:52,200 --> 00:25:55,890
gives me a histogram, Pylab dot
scatter, which gives me a

369
00:25:55,890 --> 00:25:57,970
scatter plot.

370
00:25:57,970 --> 00:26:01,350
I'll take 500 steps,
400 trials.

371
00:26:01,350 --> 00:26:04,600
I've done some more trials here,
so we get some smoother

372
00:26:04,600 --> 00:26:07,370
lines, better distributions.

373
00:26:07,370 --> 00:26:10,140
And then I'm going to look at
the usual drunk and ew drunk

374
00:26:10,140 --> 00:26:24,790
only, just to save some time.

375
00:26:24,790 --> 00:26:31,220
At least I think I am.

376
00:26:31,220 --> 00:26:35,280
My computer runs faster
when it's plugged in.

377
00:26:35,280 --> 00:26:37,040
All right, but we have
our figures here,

378
00:26:37,040 --> 00:26:40,360
let's look at them.

379
00:26:40,360 --> 00:26:42,960
So here's figure one,
the usual drunk.

380
00:26:42,960 --> 00:26:45,880
Well, all right, as we've
seen this 1000

381
00:26:45,880 --> 00:26:47,680
times, or at least six.

382
00:26:47,680 --> 00:26:55,000
So doing what it usually
does, not quite 20.

383
00:26:55,000 --> 00:26:56,820
This is kind of interesting.

384
00:26:56,820 --> 00:27:00,540
This is the final locations
of the 400 trials

385
00:27:00,540 --> 00:27:02,340
of the usual drunk.

386
00:27:02,340 --> 00:27:04,990
So we see some information here,
that we really couldn't

387
00:27:04,990 --> 00:27:08,000
get from the averages.

388
00:27:08,000 --> 00:27:12,560
That there are more of these
points near the middle, but

389
00:27:12,560 --> 00:27:18,200
there are certainly plenty
scattered around.

390
00:27:18,200 --> 00:27:21,880
And we get to see that, they're
pretty much symmetric

391
00:27:21,880 --> 00:27:24,760
around where we started.

392
00:27:24,760 --> 00:27:26,770
This is what we would
have hoped, right?

393
00:27:26,770 --> 00:27:31,000
If we had looked at it, and we
had seen all the points up

394
00:27:31,000 --> 00:27:34,250
here, or most of them up here,
we would have said, wait a

395
00:27:34,250 --> 00:27:38,190
minute, there's something
wrong with my code.

396
00:27:38,190 --> 00:27:40,550
This was supposed to not
be a biased walk,

397
00:27:40,550 --> 00:27:43,620
but an unbiased walk.

398
00:27:43,620 --> 00:27:47,580
And if I'd seen a bias in this
population, I should have

399
00:27:47,580 --> 00:27:50,180
gotten nervous that I had
some problems with my

400
00:27:50,180 --> 00:27:51,760
implementation.

401
00:27:51,760 --> 00:27:53,270
My simulation.

402
00:27:53,270 --> 00:27:56,920
So in addition to helping me
understand results, this gives

403
00:27:56,920 --> 00:28:00,540
me some confidence that I really
am doing an unbiased

404
00:28:00,540 --> 00:28:03,080
walk, right?

405
00:28:03,080 --> 00:28:04,820
Some over here, some
over here.

406
00:28:04,820 --> 00:28:09,000
Not exactly the same, but
close enough that I feel

407
00:28:09,000 --> 00:28:19,350
pretty comfortable.

408
00:28:19,350 --> 00:28:25,020
And if I look at the
distribution, which probably I

409
00:28:25,020 --> 00:28:27,930
could've if I worked really
hard, gotten from the scatter

410
00:28:27,930 --> 00:28:34,880
plot, but I really didn't want
to, we'll see that most of the

411
00:28:34,880 --> 00:28:39,750
drunks actually do end up pretty
close to the origin.

412
00:28:39,750 --> 00:28:43,960
But that there are a few that
are pretty far away.

413
00:28:43,960 --> 00:28:49,960
And again, it's close to
symmetric around zero.

414
00:28:49,960 --> 00:28:52,590
Which gives me a good feeling
that things are working the

415
00:28:52,590 --> 00:28:54,520
way they should.

416
00:28:54,520 --> 00:29:00,100
This, by the way, is what's
called a normal distribution.

417
00:29:00,100 --> 00:29:04,210
Because once upon a time, people
believed this was the

418
00:29:04,210 --> 00:29:07,850
way things usually happened.

419
00:29:07,850 --> 00:29:16,030
This is called either
normal or Gaussian.

420
00:29:16,030 --> 00:29:18,620
The mathematician Gauss was
one of the first people to

421
00:29:18,620 --> 00:29:20,870
write about this distribution.

422
00:29:20,870 --> 00:29:24,320
We'll come back to this later,
and among other things, we'll

423
00:29:24,320 --> 00:29:27,770
try and see whether normal
is really normal.

424
00:29:27,770 --> 00:29:29,960
But you see it a lot of times.

425
00:29:29,960 --> 00:29:34,410
And basically what it says is
that most of the values hang

426
00:29:34,410 --> 00:29:38,360
out around the average,
the middle.

427
00:29:38,360 --> 00:29:42,190
And there are a few outliers,
and as we get further and

428
00:29:42,190 --> 00:29:49,260
further out, fewer and
fewer points occur.

429
00:29:49,260 --> 00:29:52,120
So you'll see these kind of
curves, of distributions,

430
00:29:52,120 --> 00:29:53,390
occurring a lot.

431
00:29:53,390 --> 00:29:56,680
Again, we'll come back to
distributions in a week, or

432
00:29:56,680 --> 00:29:58,810
actually in a lecture or two.

433
00:29:58,810 --> 00:30:03,690
See a little bit more today,
and then more later on.

434
00:30:03,690 --> 00:30:13,800
All right, here's our east west
drunk, again as before,

435
00:30:13,800 --> 00:30:18,470
drifting off to around 18.

436
00:30:18,470 --> 00:30:23,730
Well, look at these dots.

437
00:30:23,730 --> 00:30:26,820
Well, this makes me feel, that
at least I know that the drunk

438
00:30:26,820 --> 00:30:31,250
is only moving east and west.
And again you'll see, pretty

439
00:30:31,250 --> 00:30:35,640
dense in the middle and
a few outliers.

440
00:30:35,640 --> 00:30:38,730
Managed to get a little further
east than west, but

441
00:30:38,730 --> 00:30:41,520
that happens.

442
00:30:41,520 --> 00:30:47,390
East is always more
attractive.

443
00:30:47,390 --> 00:30:49,810
What we might have guessed.

444
00:30:49,810 --> 00:30:56,410
And if we look at the
distribution, again we can

445
00:30:56,410 --> 00:30:59,480
look at it, and I'm just giving
you the east west

446
00:30:59,480 --> 00:31:07,160
values here, and again
it's about the same.

447
00:31:07,160 --> 00:31:11,940
So, what do I want you to take
away from these graphs?

448
00:31:11,940 --> 00:31:15,680
Different plots show you
different things.

449
00:31:15,680 --> 00:31:18,190
So I'm not going to try and
claim that of these three

450
00:31:18,190 --> 00:31:22,170
different plots we did, one
is the right one to do.

451
00:31:22,170 --> 00:31:24,000
What I'm going to assert
is that you want

452
00:31:24,000 --> 00:31:26,930
to do all of them.

453
00:31:26,930 --> 00:31:31,280
Try and visualize the data
in multiple ways.

454
00:31:31,280 --> 00:31:36,810
And sometimes it's much easier
to, even if different plots

455
00:31:36,810 --> 00:31:41,240
contain exactly the same
information, and in some sense

456
00:31:41,240 --> 00:31:45,500
of the scatter plot and the
histogram do contain exactly

457
00:31:45,500 --> 00:31:47,320
the same information.

458
00:31:47,320 --> 00:31:49,580
In fact, there's more
information in the scatter

459
00:31:49,580 --> 00:31:53,630
plot, in some sense, because I
see each one individually.

460
00:31:53,630 --> 00:31:57,520
But it's, for me, a
lot harder to see.

461
00:31:57,520 --> 00:32:00,090
If I'd ask you from the
scatter plot, are they

462
00:32:00,090 --> 00:32:02,560
normally distributed?

463
00:32:02,560 --> 00:32:04,920
You might have had a hard
time answering it.

464
00:32:04,920 --> 00:32:09,180
Particularly for the usual
drunk, where you just saw a

465
00:32:09,180 --> 00:32:10,760
lot of points.

466
00:32:10,760 --> 00:32:13,340
So the idea of some summarizing
it in the

467
00:32:13,340 --> 00:32:18,680
histogram makes it much easier
to make sense of the data, and

468
00:32:18,680 --> 00:32:21,600
see what you're doing, and to
try and learn the lesson

469
00:32:21,600 --> 00:32:32,260
you're trying to learn.

470
00:32:32,260 --> 00:32:37,560
OK, this makes sense
to people?

471
00:32:37,560 --> 00:32:40,700
All right, let's move on.

472
00:32:40,700 --> 00:32:48,040
Now we'll go back to this lovely
picture we had before.

473
00:32:48,040 --> 00:32:51,380
Any of you ever play the game
Chutes and Ladders?

474
00:32:51,380 --> 00:32:53,050
Raise your hand?

475
00:32:53,050 --> 00:32:57,790
OK, some things never
go out of style.

476
00:32:57,790 --> 00:33:04,070
Well, imagine your poor drunk
wandering through this board,

477
00:33:04,070 --> 00:33:09,430
and every once in a while,
because I'm kind of a, not a

478
00:33:09,430 --> 00:33:12,620
nice person, I've eliminated all
the ladders, and have only

479
00:33:12,620 --> 00:33:13,220
some chutes.

480
00:33:13,220 --> 00:33:15,980
And every once in a while
the drunk hits a

481
00:33:15,980 --> 00:33:18,770
chute and goes whoosh.

482
00:33:18,770 --> 00:33:26,560
And in fact, let's, for the
sake of simplicity here,

483
00:33:26,560 --> 00:33:32,230
assume that the chutes are
going to, every once in a

484
00:33:32,230 --> 00:33:35,970
while, the drunk will hit a bad
spot, and go right back to

485
00:33:35,970 --> 00:33:37,300
the origin.

486
00:33:37,300 --> 00:33:39,470
So this poor, say, heat-seeking
drunk who's

487
00:33:39,470 --> 00:33:44,260
trying to head south, does it
for a while and maybe gets

488
00:33:44,260 --> 00:33:48,230
zipped right back to where
he or she started.

489
00:33:48,230 --> 00:33:52,440
All right, I've now told
you I want to put

490
00:33:52,440 --> 00:33:55,740
some, make this happen.

491
00:33:55,740 --> 00:34:01,900
What part of the code do you
think I should change?

492
00:34:01,900 --> 00:34:02,400
Somebody?

493
00:34:02,400 --> 00:34:04,090
STUDENT: The field.

494
00:34:04,090 --> 00:34:11,360
PROFESSOR: The field,
absolutely.

495
00:34:11,360 --> 00:34:14,280
Those are good to throw
short distances.

496
00:34:14,280 --> 00:34:17,760
The field, right, because what
we're talking about here is

497
00:34:17,760 --> 00:34:21,420
not a property of the drunk,
or a property of the

498
00:34:21,420 --> 00:34:26,120
simulation, but a property of
the space in which the drunk

499
00:34:26,120 --> 00:34:30,000
is wandering.

500
00:34:30,000 --> 00:34:33,210
So again, we see that by
structuring the initial

501
00:34:33,210 --> 00:34:38,370
program around the natural
abstractions that occur in the

502
00:34:38,370 --> 00:34:46,470
problem, natural kinds of
changes can be localized.

503
00:34:46,470 --> 00:34:52,860
So let's go and change
the field.

504
00:34:52,860 --> 00:34:57,570
So I'm going to have the field
I had before plus this thing

505
00:34:57,570 --> 00:35:03,140
called an odd field.

506
00:35:03,140 --> 00:35:06,850
Odd field will be a
subclass of field.

507
00:35:06,850 --> 00:35:08,790
This is not odd as in
odd or even, this

508
00:35:08,790 --> 00:35:12,770
is odd as in strange.

509
00:35:12,770 --> 00:35:19,850
So I'm going to first define
where my chutes are.

510
00:35:19,850 --> 00:35:29,540
So is chute will take, will
get the coordinates, and

511
00:35:29,540 --> 00:35:32,650
assign it to x and y, so it
gets the coordinates of a

512
00:35:32,650 --> 00:35:37,220
place, and then will return
abs x minus abs

513
00:35:37,220 --> 00:35:39,680
y is equal to zero.

514
00:35:39,680 --> 00:35:45,780
So, where are my chutes
going to occur?

515
00:35:45,780 --> 00:35:53,980
Somebody?

516
00:35:53,980 --> 00:36:00,130
Yeah, kind of radiating
out from the origin.

517
00:36:00,130 --> 00:36:06,720
Any place, so I'll have
my origin, here.

518
00:36:06,720 --> 00:36:08,200
Got my graph.

519
00:36:08,200 --> 00:36:13,860
And all along here, any place
x and y are equal,

520
00:36:13,860 --> 00:36:18,180
there'll be a chute.

521
00:36:18,180 --> 00:36:21,590
Now, I have to be a little bit
careful that it doesn't happen

522
00:36:21,590 --> 00:36:27,110
here, or we won't
get anywhere.

523
00:36:27,110 --> 00:36:32,400
So, and then move, in odd field,
will call field dot

524
00:36:32,400 --> 00:36:38,530
move, which is unchanged from
what it was before.

525
00:36:38,530 --> 00:36:43,130
And then it will say, if self
dot is chute, set the location

526
00:36:43,130 --> 00:36:45,690
back to 0,0.

527
00:36:45,690 --> 00:36:50,120
So the poor drunk will move as
before, and if he or she hits

528
00:36:50,120 --> 00:36:58,220
this wormhole, get instantly
teleported back to the origin.

529
00:36:58,220 --> 00:37:01,910
Very small change, but it will
give us rather different

530
00:37:01,910 --> 00:37:08,410
behaviors, at least
I think it will.

531
00:37:08,410 --> 00:37:17,460
And then if we look at how we
use it, not much changes.

532
00:37:17,460 --> 00:37:20,670
Except, what do you think is
going to have to change?

533
00:37:20,670 --> 00:37:23,420
Where will I have to make
the change, somebody?

534
00:37:23,420 --> 00:37:27,840
You can actually see
the code here.

535
00:37:27,840 --> 00:37:31,710
I'll have to get a different
field, right?

536
00:37:31,710 --> 00:37:35,880
So, at the place where I
instantiate a field, instead

537
00:37:35,880 --> 00:37:40,600
of instantiating a field, I'll
instantiate an odd field.

538
00:37:40,600 --> 00:37:44,130
Now if I'd had 16 different
kinds of odd fields, or even

539
00:37:44,130 --> 00:37:49,500
two, I might well have done
what I did with drunk.

540
00:37:49,500 --> 00:37:53,860
But in order to sort of minimize
things, I've done

541
00:37:53,860 --> 00:37:55,040
something much smaller.

542
00:37:55,040 --> 00:37:57,400
So let's see, where did
we get the field?

543
00:37:57,400 --> 00:37:59,780
I don't even remember.

544
00:37:59,780 --> 00:38:03,140
So, here's what I do when I'm
looking at my program and I

545
00:38:03,140 --> 00:38:06,530
want to see where
something is.

546
00:38:06,530 --> 00:38:13,090
I use the text editor, and I'm
going to search for it.

547
00:38:13,090 --> 00:38:24,500
Let's see.

548
00:38:24,500 --> 00:38:31,320
Well, it's not there.

549
00:38:31,320 --> 00:38:33,710
There it is.

550
00:38:33,710 --> 00:38:37,280
So here when I get my field, I
get my drunk, and then I get a

551
00:38:37,280 --> 00:38:44,280
field which is an odd field.

552
00:38:44,280 --> 00:38:46,460
All right, anyone want to
speculate what will happen

553
00:38:46,460 --> 00:38:54,170
when I run this one?

554
00:38:54,170 --> 00:38:59,150
Think the drunk will be closer
or further from the origin

555
00:38:59,150 --> 00:39:00,870
when we're done?

556
00:39:00,870 --> 00:39:03,050
Who thinks closer?

557
00:39:03,050 --> 00:39:05,120
Who thinks further?

558
00:39:05,120 --> 00:39:07,720
Who thinks no difference?

559
00:39:07,720 --> 00:39:09,920
Well, you're right, I mean it's
sort of logical if we

560
00:39:09,920 --> 00:39:12,180
think about it, that if every
once in a while you get to

561
00:39:12,180 --> 00:39:15,220
zipped, back it's going to be
harder to get further away.

562
00:39:15,220 --> 00:39:17,890
We'll see different behaviors,
by the way, it depends on the

563
00:39:17,890 --> 00:39:19,200
drunk, right?

564
00:39:19,200 --> 00:39:23,832
That's true for the usual drunk,
but maybe not for the

565
00:39:23,832 --> 00:39:27,740
east west drunk.

566
00:39:27,740 --> 00:39:29,590
We can see what happens.

567
00:39:29,590 --> 00:39:33,900
So let's try some.

568
00:39:33,900 --> 00:39:36,030
We'll do it for the usual
drunk, which is the more

569
00:39:36,030 --> 00:39:58,030
interesting case
to start with.

570
00:39:58,030 --> 00:40:01,670
So you'll see whereas before we
were up just shy of 20 most

571
00:40:01,670 --> 00:40:06,140
of the time, here we don't get
very much much, further.

572
00:40:06,140 --> 00:40:13,040
We do get steadily further,
but at a lot slower pace.

573
00:40:13,040 --> 00:40:16,220
What do you think it means that,
before you remember, we

574
00:40:16,220 --> 00:40:17,830
saw a pretty smooth line.

575
00:40:17,830 --> 00:40:21,200
Here we see what may look you
like a fat line, but is in

576
00:40:21,200 --> 00:40:25,660
fact a line with a lot
of wiggles in it.

577
00:40:25,660 --> 00:40:30,680
What does that imply, about
what's going on in the

578
00:40:30,680 --> 00:40:36,430
simulation, the fact that line
is jagged rather than smooth

579
00:40:36,430 --> 00:40:41,490
as it was before?

580
00:40:41,490 --> 00:40:43,390
Pardon?

581
00:40:43,390 --> 00:40:44,070
Can't hear you?

582
00:40:44,070 --> 00:40:44,470
STUDENT: That it jumps.

583
00:40:44,470 --> 00:40:50,320
PROFESSOR: Well, certainly it
happens because of the jumps,

584
00:40:50,320 --> 00:40:51,680
but what else is going on?

585
00:40:51,680 --> 00:40:55,910
But what is it, sort of, imply
more, whoa, sorry about that,

586
00:40:55,910 --> 00:40:57,410
more generally?

587
00:40:57,410 --> 00:41:01,190
Right, what it's saying, is
that because of the jumps,

588
00:41:01,190 --> 00:41:07,190
there's a lot more variation
from trial to trial.

589
00:41:07,190 --> 00:41:09,860
Not surprising, right, because
you sometimes you hit these

590
00:41:09,860 --> 00:41:12,480
wormholes and sometimes
you don't.

591
00:41:12,480 --> 00:41:15,330
So let's look at this
other figure.

592
00:41:15,330 --> 00:41:18,460
This is kind of interesting.

593
00:41:18,460 --> 00:41:24,090
So here's the scatter plot.

594
00:41:24,090 --> 00:41:37,070
And if we zoom in on it, what
we see here is, there are

595
00:41:37,070 --> 00:41:41,710
essentially holes, you
know, these alleys

596
00:41:41,710 --> 00:41:46,410
where no points lie.

597
00:41:46,410 --> 00:41:50,380
Just as we might have
guessed from here.

598
00:41:50,380 --> 00:41:52,940
The only way you would get a
point lying on these alleys,

599
00:41:52,940 --> 00:41:55,640
if it happened to be the very
last step of the simulation,

600
00:41:55,640 --> 00:41:58,740
not even then, because the last
thing we do is go back to

601
00:41:58,740 --> 00:42:00,540
the origin.

602
00:42:00,540 --> 00:42:04,840
So we can look at this, and say,
well, sure enough, this

603
00:42:04,840 --> 00:42:15,790
is keeping people off of a
certain part of the field.

604
00:42:15,790 --> 00:42:20,110
All right, some of you will be
happy to know we are, for the

605
00:42:20,110 --> 00:42:22,590
moment, actually maybe for the
whole rest of the term,

606
00:42:22,590 --> 00:42:24,010
leaving the notion of drunks.

607
00:42:24,010 --> 00:42:26,880
Though not of random walks.

608
00:42:26,880 --> 00:42:30,260
All right, any questions about
what's going on here?

609
00:42:30,260 --> 00:42:32,970
And again, I would urge you to
sort of study the code and the

610
00:42:32,970 --> 00:42:35,910
way it's been factored to
accomplish these things.

611
00:42:35,910 --> 00:42:38,370
And play with it, and look
at what you get from the

612
00:42:38,370 --> 00:42:40,870
different plots.

613
00:42:40,870 --> 00:42:44,620
I now want to pull back from
this specific example, and

614
00:42:44,620 --> 00:42:50,310
spend a few minutes talking
about simulation in general.

615
00:42:50,310 --> 00:42:55,720
Computer simulation really grew
hand in hand with the

616
00:42:55,720 --> 00:42:59,400
development of the computers,
from the very beginning.

617
00:42:59,400 --> 00:43:02,520
The first large-scale deployment
of computer

618
00:43:02,520 --> 00:43:15,720
simulation was as part of
the Manhattan Project.

619
00:43:15,720 --> 00:43:19,600
This was done during the war
to model the process of

620
00:43:19,600 --> 00:43:29,380
nuclear detonation.

621
00:43:29,380 --> 00:43:33,840
And, in fact, what they did was
a simulation of 12 hard

622
00:43:33,840 --> 00:43:37,110
spheres, and what would
happen when they would

623
00:43:37,110 --> 00:43:38,960
bump into each other.

624
00:43:38,960 --> 00:43:43,300
It was a huge step forward to
do it with simulation, since

625
00:43:43,300 --> 00:43:47,110
the whole project had been
stalled by their attempt to do

626
00:43:47,110 --> 00:43:48,830
this analytically.

627
00:43:48,830 --> 00:43:51,050
They were unable to actually
solve the problem

628
00:43:51,050 --> 00:43:52,480
analytically.

629
00:43:52,480 --> 00:43:55,120
And it was only when they hit
upon the notion of using a

630
00:43:55,120 --> 00:43:58,610
computer, a relatively new
tool in those days, to

631
00:43:58,610 --> 00:44:00,690
simulate it, that they
were able to get the

632
00:44:00,690 --> 00:44:02,700
answers they needed.

633
00:44:02,700 --> 00:44:06,910
And they did it using something
called a Monte Carlo

634
00:44:06,910 --> 00:44:19,180
simulation.

635
00:44:19,180 --> 00:44:22,010
Now in fact, that's just what
we've been doing with the

636
00:44:22,010 --> 00:44:24,960
random walk, the Monte Carlo
simulation, and I'll come back

637
00:44:24,960 --> 00:44:27,160
to that in a minute.

638
00:44:27,160 --> 00:44:32,940
In general, the thing to think
about, is we use simulation

639
00:44:32,940 --> 00:44:39,500
when we really can't get a
closed form analytic solution.

640
00:44:39,500 --> 00:44:42,750
If you can put down a system of
equations and easily solve

641
00:44:42,750 --> 00:44:47,510
it to get an answer, that's
usually the right thing to do.

642
00:44:47,510 --> 00:44:50,780
But when we can't easily do
that, the right thing to do is

643
00:44:50,780 --> 00:44:54,940
typically to fall back
on simulation.

644
00:44:54,940 --> 00:44:57,930
Sometimes even when we can do
it analytically, simulation

645
00:44:57,930 --> 00:45:05,340
has some advantages, as we'll
be seeing as we go forward.

646
00:45:05,340 --> 00:45:07,970
What we're typically doing
when we're simulating

647
00:45:07,970 --> 00:45:31,960
anything, is we're attempting
to generate a sample of

648
00:45:31,960 --> 00:45:43,320
representative scenarios.

649
00:45:43,320 --> 00:45:47,010
Because an exhaustive
enumeration of all possible

650
00:45:47,010 --> 00:45:50,330
states would be impossible.

651
00:45:50,330 --> 00:45:53,550
So again, if sometimes you can't
solve it analytically,

652
00:45:53,550 --> 00:45:57,530
you can exhaustively enumerate
the space and then see

653
00:45:57,530 --> 00:45:59,280
what's going on.

654
00:45:59,280 --> 00:46:01,210
But again, usually you can't.

655
00:46:01,210 --> 00:46:05,150
And so we look for a sample, and
the key is, it's gotta be

656
00:46:05,150 --> 00:46:07,960
representative.

657
00:46:07,960 --> 00:46:11,570
Of what we would
get in reality.

658
00:46:11,570 --> 00:46:14,500
We'll see an example
of that shortly.

659
00:46:14,500 --> 00:46:20,120
So simulation attempts to build
an experimental device

660
00:46:20,120 --> 00:46:25,890
that will act like the real
system in important aspects.

661
00:46:25,890 --> 00:46:29,170
So I always think of
a simulation as an

662
00:46:29,170 --> 00:46:32,180
experimental device.

663
00:46:32,180 --> 00:46:37,440
And every time I run it, I'm
running an experiment designed

664
00:46:37,440 --> 00:46:41,430
to give me some information
about the real world.

665
00:46:41,430 --> 00:46:45,110
These things are enormously
popular, so if you were to get

666
00:46:45,110 --> 00:46:52,690
on Google and look
at simulation.

667
00:46:52,690 --> 00:47:00,540
So for, example, we could Google
simulation finance.

668
00:47:00,540 --> 00:47:06,230
You know, we see we get
about 3,000,000 hits.

669
00:47:06,230 --> 00:47:13,600
We can do biology.

670
00:47:13,600 --> 00:47:16,360
We get twice as many hits,
showing the relative

671
00:47:16,360 --> 00:47:21,770
importance in the world of
biology and finance.

672
00:47:21,770 --> 00:47:25,150
For all you Course 7 majors,
we should make sure that

673
00:47:25,150 --> 00:47:27,650
should compare biology
to physics.

674
00:47:27,650 --> 00:47:31,280
Oh dear, we'll see that physics
is even more important

675
00:47:31,280 --> 00:47:35,170
than biology.

676
00:47:35,170 --> 00:47:38,320
And I know we have a lot of
Course 2 students in here, so

677
00:47:38,320 --> 00:47:43,310
let's try mechanical.

678
00:47:43,310 --> 00:47:45,520
Oh, lot of those, too.

679
00:47:45,520 --> 00:47:48,680
And if I did baseball, or
football, we'd get more than

680
00:47:48,680 --> 00:47:50,720
any of them.

681
00:47:50,720 --> 00:47:54,110
Showing the real importance
of things in the world.

682
00:47:54,110 --> 00:48:00,570
As we look at simulations, I
want you to keep in mind that

683
00:48:00,570 --> 00:48:12,930
they are typically descriptive
not prescriptive.

684
00:48:12,930 --> 00:48:18,230
So by that I mean, they describe
a situation, they

685
00:48:18,230 --> 00:48:22,590
don't tell you what
the answer is.

686
00:48:22,590 --> 00:48:28,790
So another way to think about
this is, a simulation is not

687
00:48:28,790 --> 00:48:31,370
an optimization procedure.

688
00:48:31,370 --> 00:48:35,180
When we looked at optimization,
that was

689
00:48:35,180 --> 00:48:35,840
prescriptive.

690
00:48:35,840 --> 00:48:41,280
We ran an optimization
algorithm, and it gave us the

691
00:48:41,280 --> 00:48:45,010
best possible solution.

692
00:48:45,010 --> 00:48:48,420
A simulation typically doesn't
give you this best possible

693
00:48:48,420 --> 00:48:53,150
solution, but if you give it
the starting point, it will

694
00:48:53,150 --> 00:48:58,780
tell you the consequences
of that.

695
00:48:58,780 --> 00:49:01,930
Now, can we use simulation
to do optimization?

696
00:49:01,930 --> 00:49:03,780
Absolutely.

697
00:49:03,780 --> 00:49:07,440
For example, we can use it
to do guess and check.

698
00:49:07,440 --> 00:49:10,510
Where we guess a, probably not
to get a truly optimal

699
00:49:10,510 --> 00:49:15,950
solution, but to get a good
solution, we can guess various

700
00:49:15,950 --> 00:49:18,880
possibilities, simulate
them, and see.

701
00:49:18,880 --> 00:49:21,750
People do that all the time.

702
00:49:21,750 --> 00:49:27,120
If they want to see what's the
right number of checkout

703
00:49:27,120 --> 00:49:30,920
counters at the supermarket,
or what's the right airline

704
00:49:30,920 --> 00:49:32,540
schedule to use?

705
00:49:32,540 --> 00:49:35,803
They'll make a guess, they'll
simulate it, and they'll say

706
00:49:35,803 --> 00:49:39,980
all right, this was better than
this, so we'll choose it.

707
00:49:39,980 --> 00:49:42,170
All right, I'm going
to stop here.

708
00:49:42,170 --> 00:49:45,920
Next time we're going to get
more deeply into the actual

709
00:49:45,920 --> 00:49:49,720
stochastics, the probability
distributions, and start

710
00:49:49,720 --> 00:49:52,580
understanding what's going
on under the covers.